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Envelopes and refinements are examples of colimits and limits respectively, which play an important role in Functional analysis (specifically, in the theory of topological algebras).

A typical example is the Arens-Michael envelope. Intuitively it is an operation (a functor) that turns each topological algebra $A$ into its nearest from the outside holomorphic algebra $\operatorname{Env}A$. It can be defined as the colimit of the system of Banach quotient algebras of $A$: $$ \operatorname{Env}A=\lim_{\leftarrow} A/U, $$ where $U$ means a submultiplicative closed convex balanced neighborhood of zero in $A$ $$ U\cdot U\subseteq U, $$ and $$ A/U=(A/\bigcap_{\varepsilon>0}\varepsilon\cdot U)^\blacktriangledown $$ is the completion of the quotient algebra $A/\bigcap_{\varepsilon>0}\varepsilon\cdot U$ endowed with the norm topology with the unit ball $U+\bigcap_{\varepsilon>0}\varepsilon\cdot U$ (the space $A/U$ will be a Banach algebra with respect to the "multiplication inherited from $A$").

As an illustration, if $A$ is the algebra ${\mathcal P}(M)$ of polynomials on an affine algebraic manifold $M$, then its Arens-Michael envelope $\text{Env}A$ is exactly the algebra ${\mathcal O}(M)$ of holomorphic functions on $M$: $$ \operatorname{Env}{\mathcal P}(M)={\mathcal O}(M). $$
The non-commutative case is especially interesting, since it generates non-trivial computations in non-commutative geometry. For example, the "polynomial version" of the so-called $az+b$-group is turned into its "holomorphic version": $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}_q({\mathbb C}^\times\ltimes{\mathbb C}), $$
for $|q|=1$, but the formula becomes different in the case of $|q|\ne 1$: $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}({\mathbb C}^\times)\underset{\delta^q}{\overset{z}{\odot}}{\mathcal P}^\star({\mathbb C}) $$
(see details in my paper). Alexei Pirkovskii, who reanimated this topic after some oblivion, calculates the Arens-Michael envelopes of different algebras as exercises.

Similarly, there are natural functors of taking

• the nearest continuous topological algebra (see also here) with the formula $$ \operatorname{Env}A={\mathcal C}(M) $$
  for a subalgebra $A$ in ${\mathcal C}(M)$ having $M$ as the involutive spectrum, and

• the nearest smooth topological algebra with the formula $$ \operatorname{Env}A={\mathcal C}^\infty(M) $$
  for a subalgebra $A$ in ${\mathcal C}^\infty(M)$ having $M$ as the involutive spectrum and $T_x(M)$ as the involutive tangent space in each point $x\in M$.

Envelopes and refinements are often colimits and limits respectively, and they play an important role in Functional analysis (specifically, in the theory of topological algebras).

A typical example is the Arens-Michael envelope. Intuitively it is an operation (a functor) that turns each topological algebra $A$ into its nearest from the outside holomorphic algebra $\operatorname{Env}A$. It can be defined as the colimit of the system of Banach quotient algebras of $A$: $$ \operatorname{Env}A=\lim_{\leftarrow} A/U, $$ where $U$ means a submultiplicative closed convex balanced neighborhood of zero in $A$ $$ U\cdot U\subseteq U, $$ and $$ A/U=(A/\bigcap_{\varepsilon>0}\varepsilon\cdot U)^\blacktriangledown $$ is the completion of the quotient algebra $A/\bigcap_{\varepsilon>0}\varepsilon\cdot U$ endowed with the norm topology with the unit ball $U+\bigcap_{\varepsilon>0}\varepsilon\cdot U$ (the space $A/U$ will be a Banach algebra with respect to the "multiplication inherited from $A$").

As an illustration, if $A$ is the algebra ${\mathcal P}(M)$ of polynomials on an affine algebraic manifold $M$, then its Arens-Michael envelope $\text{Env}A$ is exactly the algebra ${\mathcal O}(M)$ of holomorphic functions on $M$: $$ \operatorname{Env}{\mathcal P}(M)={\mathcal O}(M). $$
The non-commutative case is especially interesting, since it generates non-trivial computations in non-commutative geometry. For example, the "polynomial version" of the so-called $az+b$-group is turned into its "holomorphic version": $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}_q({\mathbb C}^\times\ltimes{\mathbb C}), $$
for $|q|=1$, but the formula becomes different in the case of $|q|\ne 1$: $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}({\mathbb C}^\times)\underset{\delta^q}{\overset{z}{\odot}}{\mathcal P}^\star({\mathbb C}) $$
(see details in my paper). Alexei Pirkovskii, who reanimated this topic after some oblivion, calculates the Arens-Michael envelopes of different algebras as exercises.

Similarly, there are natural functors of taking

• the nearest continuous topological algebra (see also here) with the formula $$ \operatorname{Env}A={\mathcal C}(M) $$
  for a subalgebra $A$ in ${\mathcal C}(M)$ having $M$ as the involutive spectrum, and

• the nearest smooth topological algebra with the formula $$ \operatorname{Env}A={\mathcal C}^\infty(M) $$
  for a subalgebra $A$ in ${\mathcal C}^\infty(M)$ having $M$ as the involutive spectrum and $T_x(M)$ as the involutive tangent space in each point $x\in M$.

Envelopes and refinements are examples of colimits and limits respectively, which play an important role in Functional analysis (specifically, in the theory of topological algebras).

The Arens-Michael envelope is intuitively an operation (a functor) that turns each topological algebra $A$ into its nearest from the outside holomorphic algebra $\operatorname{Env}A$. It can be defined as the colimit of the system of Banach quotient algebras of $A$: $$ \operatorname{Env}A=\lim_{\leftarrow} A/U, $$ where $U$ means a submultiplicative closed convex balanced neighborhood of zero in $A$ $$ U\cdot U\subseteq U, $$ and $$ A/U=(A/\bigcap_{\varepsilon>0}\varepsilon\cdot U)^\blacktriangledown $$ is the completion of the quotient algebra $A/\bigcap_{\varepsilon>0}\varepsilon\cdot U$ endowed with the norm topology with the unit ball $U+\bigcap_{\varepsilon>0}\varepsilon\cdot U$ (the space $A/U$ will be a Banach algebra with respect to the "multiplication inherited from $A$").

As an illustration, if $A$ is the algebra ${\mathcal P}(M)$ of polynomials on an affine algebraic manifold $M$, then its Arens-Michael envelope $\text{Env}A$ is exactly the algebra ${\mathcal O}(M)$ of holomorphic functions on $M$: $$ \operatorname{Env}{\mathcal P}(M)={\mathcal O}(M). $$
The non-commutative case is especially interesting, since it generates non-trivial computations in non-commutative geometry. For example, the "polynomial version" of the so-called $az+b$-group is turned into its "holomorphic version": $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}_q({\mathbb C}^\times\ltimes{\mathbb C}), $$
for $|q|=1$, but the formula becomes different in the case of $|q|\ne 1$: $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}({\mathbb C}^\times)\underset{\delta^q}{\overset{z}{\odot}}{\mathcal P}^\star({\mathbb C}) $$
(see details in my paper). Alexei Pirkovskii, who reanimated this topic after some oblivion, calculates the Arens-Michael envelopes of different algebras as exercises.

Similarly, there are natural functors of taking

• the nearest continuous topological algebra (see also here) with the formula $$ \operatorname{Env}A={\mathcal C}(M) $$
  for a subalgebra $A$ in ${\mathcal C}(M)$ having $M$ as the involutive spectrum, and

• the nearest smooth topological algebra with the formula $$ \operatorname{Env}A={\mathcal C}^\infty(M) $$
  for a subalgebra $A$ in ${\mathcal C}^\infty(M)$ having $M$ as the involutive spectrum and $T_x(M)$ as the involutive tangent space in each point $x\in M$.

Envelopes and refinements are examples of colimits and limits respectively, which play an important role in Functional analysis (specifically, in the theory of topological algebras).

A typical example is the Arens-Michael envelope. Intuitively it is an operation (a functor) that turns each topological algebra $A$ into its nearest from the outside holomorphic algebra $\operatorname{Env}A$. It can be defined as the colimit of the system of Banach quotient algebras of $A$: $$ \operatorname{Env}A=\lim_{\leftarrow} A/U, $$ where $U$ means a submultiplicative closed convex balanced neighborhood of zero in $A$ $$ U\cdot U\subseteq U, $$ and $$ A/U=(A/\bigcap_{\varepsilon>0}\varepsilon\cdot U)^\blacktriangledown $$ is the completion of the quotient algebra $A/\bigcap_{\varepsilon>0}\varepsilon\cdot U$ endowed with the norm topology with the unit ball $U+\bigcap_{\varepsilon>0}\varepsilon\cdot U$ (the space $A/U$ will be a Banach algebra with respect to the "multiplication inherited from $A$").

As an illustration, if $A$ is the algebra ${\mathcal P}(M)$ of polynomials on an affine algebraic manifold $M$, then its Arens-Michael envelope $\text{Env}A$ is exactly the algebra ${\mathcal O}(M)$ of holomorphic functions on $M$: $$ \operatorname{Env}{\mathcal P}(M)={\mathcal O}(M). $$
The non-commutative case is especially interesting, since it generates non-trivial computations in non-commutative geometry. For example, the "polynomial version" of the so-called $az+b$-group is turned into its "holomorphic version": $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}_q({\mathbb C}^\times\ltimes{\mathbb C}), $$
for $|q|=1$, but the formula becomes different in the case of $|q|\ne 1$: $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}({\mathbb C}^\times)\underset{\delta^q}{\overset{z}{\odot}}{\mathcal P}^\star({\mathbb C}) $$
(see details in my paper). Alexei Pirkovskii, who reanimated this topic after some oblivion, calculates the Arens-Michael envelopes of different algebras as exercises.

Similarly, there are natural functors of taking

• the nearest continuous topological algebra (see also here) with the formula $$ \operatorname{Env}A={\mathcal C}(M) $$
  for a subalgebra $A$ in ${\mathcal C}(M)$ having $M$ as the involutive spectrum, and

• the nearest smooth topological algebra with the formula $$ \operatorname{Env}A={\mathcal C}^\infty(M) $$
  for a subalgebra $A$ in ${\mathcal C}^\infty(M)$ having $M$ as the involutive spectrum and $T_x(M)$ as the involutive tangent space in each point $x\in M$.

Envelopes and refinements are examples of colimits and limits respectively that play important role in Functional analysis (specifically, in the theory of topological algebras).

The Arens-Michael envelope is intuitively an operation (a functor) that turns each topological algebra $A$ into its nearest from the outside holomorphic algebra $\operatorname{Env}A$. It can be defined as the colimit of the system of Banach quotient algebras of $A$: $$ \operatorname{Env}A=\lim_{\leftarrow} A/U, $$ where $U$ means a submultiplicative closed convex balanced neighborhood of zero in $A$ $$ U\cdot U\subseteq U, $$ and $$ A/U=(A/\bigcap_{\varepsilon>0}\varepsilon\cdot U)^\blacktriangledown $$ is the completion of the quotient algebra $A/\bigcap_{\varepsilon>0}\varepsilon\cdot U$ endowed with the norm topology with the unit ball $U+\bigcap_{\varepsilon>0}\varepsilon\cdot U$ (the space $A/U$ will be a Banach algebra with respect to the "multiplication inherited from $A$").

As an illustration, if $A$ is the algebra ${\mathcal P}(M)$ of polynomials on an affine algebraic manifold $M$, then its Arens-Michael envelope $\text{Env}A$ is exactly the algebra ${\mathcal O}(M)$ of holomorphic functions on $M$: $$ \operatorname{Env}{\mathcal P}(M)={\mathcal O}(M). $$
The non-commutative case is especially interesting, since it generates non-trivial computations in non-commutative geometry. For example, the "polynomial version" of the so-called $az+b$-group is turned into its "holomorphic version": $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}_q({\mathbb C}^\times\ltimes{\mathbb C}), $$
for $|q|=1$, but the formula becomes different in the case of $|q|\ne 1$: $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}({\mathbb C}^\times)\underset{\delta^q}{\overset{z}{\odot}}{\mathcal P}^\star({\mathbb C}) $$
(see details in my paper). Alexei Pirkovskii, who reanimated this topic after some oblivion, calculates the Arens-Michael envelopes of different algebras as exercises.

Similarly, there are natural functors of taking

• the nearest continuous topological algebra (see also here) with the formula $$ \operatorname{Env}A={\mathcal C}(M) $$
  for a subalgebra $A$ in ${\mathcal C}(M)$ having $M$ as the involutive spectrum, and

• the nearest smooth topological algebra with the formula $$ \operatorname{Env}A={\mathcal C}^\infty(M) $$
  for a subalgebra $A$ in ${\mathcal C}^\infty(M)$ having $M$ as the involutive spectrum and $T_x(M)$ as the involutive tangent space in each point $x\in M$.

Envelopes and refinements are examples of colimits and limits respectively, which play an important role in Functional analysis (specifically, in the theory of topological algebras).

The Arens-Michael envelope is intuitively an operation (a functor) that turns each topological algebra $A$ into its nearest from the outside holomorphic algebra $\operatorname{Env}A$. It can be defined as the colimit of the system of Banach quotient algebras of $A$: $$ \operatorname{Env}A=\lim_{\leftarrow} A/U, $$ where $U$ means a submultiplicative closed convex balanced neighborhood of zero in $A$ $$ U\cdot U\subseteq U, $$ and $$ A/U=(A/\bigcap_{\varepsilon>0}\varepsilon\cdot U)^\blacktriangledown $$ is the completion of the quotient algebra $A/\bigcap_{\varepsilon>0}\varepsilon\cdot U$ endowed with the norm topology with the unit ball $U+\bigcap_{\varepsilon>0}\varepsilon\cdot U$ (the space $A/U$ will be a Banach algebra with respect to the "multiplication inherited from $A$").

As an illustration, if $A$ is the algebra ${\mathcal P}(M)$ of polynomials on an affine algebraic manifold $M$, then its Arens-Michael envelope $\text{Env}A$ is exactly the algebra ${\mathcal O}(M)$ of holomorphic functions on $M$: $$ \operatorname{Env}{\mathcal P}(M)={\mathcal O}(M). $$
The non-commutative case is especially interesting, since it generates non-trivial computations in non-commutative geometry. For example, the "polynomial version" of the so-called $az+b$-group is turned into its "holomorphic version": $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}_q({\mathbb C}^\times\ltimes{\mathbb C}), $$
for $|q|=1$, but the formula becomes different in the case of $|q|\ne 1$: $$ \operatorname{Env}{\mathcal P}_q({\mathbb C}^\times\ltimes{\mathbb C})={\mathcal O}({\mathbb C}^\times)\underset{\delta^q}{\overset{z}{\odot}}{\mathcal P}^\star({\mathbb C}) $$
(see details in my paper). Alexei Pirkovskii, who reanimated this topic after some oblivion, calculates the Arens-Michael envelopes of different algebras as exercises.

Similarly, there are natural functors of taking

• the nearest continuous topological algebra (see also here) with the formula $$ \operatorname{Env}A={\mathcal C}(M) $$
  for a subalgebra $A$ in ${\mathcal C}(M)$ having $M$ as the involutive spectrum, and

• the nearest smooth topological algebra with the formula $$ \operatorname{Env}A={\mathcal C}^\infty(M) $$
  for a subalgebra $A$ in ${\mathcal C}^\infty(M)$ having $M$ as the involutive spectrum and $T_x(M)$ as the involutive tangent space in each point $x\in M$.